This was an interesting class. I assigned the first portfolio problem. These portfolio problems are designed to have students fully explain their reasoning and justify their results. I allow students to make revisions on their work until the get the results correct. These portfolio problems become mentor text that shows students what they need to do when responding to questions. Students do not receive credit for a portfolio problem until it is 100% correct.
I also use a grading standard of Essential Correct (E), Partially Correct (P), and Incomplete (I). An E says that students "get it" and know what they are doing and can communicate their results and reasoning. There may be minor issues but that these are things that a student could, in essence, self-correct. A P indicates a student knows what they are doing but gets stuck, does not explain their reasoning, or would need some prodding or help through questioning to move on. An I indicates the student cannot proceed without a lot of assistance or provides an answer without any explanation of their thinking or justification of their results.
All graded work is scored using E, P, and I. On a test, a student getting all P's would receive a grade of C. A student getting all E's would receive an A. A student receiving a mix of E's and P's would fall somewhere in between.
After going through the portfolio problem requirements and grading, we started on the day's work. This involved using values in a table to explore functional relationship. We started with linear equations. I had students create linear tables and look at the difference in the y-value. Students quickly saw that the difference was equal to the coefficient. After confirming this worked for other linear equations I posed the question as to why this would happen. Some students related it to slope but didn't come up with any definitive response. I left this as an open question and then asked if this result would always work.
A student wondered if it would work with quadratic equations and so we explored the situation for two specific equations. They saw that the first difference wasn't a constant but that if you took the second difference the result was. They also noticed that it looked like you should multiply the coefficient times the exponent. We explored this for a couple of more quadratics.
The question again was would this hold and we proceeded to explore this for cubic equations. Students found that the third difference was constant and the third difference looked like (coefficient) x (exponent) x 2. Students were stumped about what there was a 2 being multiplied.
I recapped what we learned so far, that linear equations had a first difference that was constant, quadratic equations had a second difference that was constant, and that cubic equations had a third difference that was constant. We also knew that the difference took on the form of (coefficient) x (exponent) x (something) where the something was a value of 2 for cubic equations.
I asked the class what this would mean for a quartic equation and they said that the fourth difference should be a constant. We used y = x4 to verify this. Students confirmed that the fourth difference was constant and saw that its value was 24. I then referenced our other result of 24 = (coefficient) x (exponent) x (something) = 1 x 4 x (something) = 1 x 4 x 6.
Several students noticed that 6 was the (exponent) x (something) component of the cubic equation and suggested writing out the value as 24 = 1 x 4 x 3 x 2. At this point someone suggested to write a one at the end of the sequence and there were several aha's that the coefficient was being multiplied by the factorial of the exponent, i.e. 24 = 1 x 4!.
I then explained that if a polynomial equation had as its largest term anxn then the nth difference would be
an x n!.
We then looked at the first few pentagonal numbers to see what we could tell about the equation. The values we had were 1, 5, 12, and 22. Students saw that the second difference was a constant and its value was 3. There was some confusion as to whether that meant it was a quadratic or cubic. I said that the second difference was 2 and asked what this meant. Most came to the realization that this meant it had to be a quadratic equation. I then asked what the coefficient was. Again, the fact that the coefficient was a fraction threw them off a bit. I wrote out that a x 2! = 3 and asked what a had to equal. They said it had to be 1.5 or 3/2. I then said that this meant the equation for pentagonal numbers started with the term 1.5x2.
I created a table and talked about the contribution that 1.5x2 made to the y-value. We subtracted this contribution out and saw that the results decreased by .5 each time. This meant we had a linear component of the form -.5x. Repeating the process we saw that the result was always 0. The provided an equation for pentagonal numbers of: Pn = 1.5x2 - .5x. This can be written as Pn = 3x(x-1)/2 the more traditional form of the equation.
I then had students summarize what they had learned about taking differences and how these results could be used to find equations.
Visit the summary page for a student's perspective on the class and to view the lesson slides.
Thursday, January 17, 2013
Wednesday, January 16, 2013
IPS - Day 5
Today we transitioned from exploring random events to simulation.
To connect with the idea of what randomness looks like and the Law of Large numbers the class started with a couple of quick assessments of understanding. Students voted on which heads-tails sequence was more likely. Roughly half thought the probability was the same and about 25% went to each individual sequence. The thoughts around the "hot hand" concept was divided roughly the same, half thought there was no such thing, 25% felt like the person was likely to continue with their streak, and 25% thought the streak would end. We discussed the reasoning and students were able to articulate the idea of the Law of Large Numbers.
The One Boy Family Planning activity comes from NTCM's Navigating Through Probability in Grades 9-12. I find the activities in this book to be enlightening and accessible to all levels of students.
I start the activity by asking students to read through the scenario and then develop a hypothesis as to what they think will happen given their current knowledge. We then briefly discuss the simulation aspect of the problem and I ask them to consider other ways that the situation could be simulated. Students came up with flipping a coin, using the random number generator, and mentioned that any device that allowed for two equal outcomes would work.
Students then proceeded to simulate the situation and record their results. Some groups will have misunderstanding of the simulation process so it's a good idea to walk around and make sure everyone is on track.
I ask students to hold off completing the second table as this usually causes problems. I actually have students create two columns: Total Children and Cummulative Children. I give a brief example to illustrate what this looks like. Again, most will get it but some may wander off track. I have students the total number of children and divide it by the number of households and ask what this represents. After a little thought the come up with the average children per household. I then explain how to use the second table to calculate their probabilities.
We discussed their expectations and surprises that came out of the investigation. Many students were somewhat surprised by the average children per household but they did anticipate that roughly half the houses would have a single child since the probability of having a boy was 50%.
I gave students a few minutes to collect their thoughts and summarize their thinking about randomness and to write two questions that they have. I will start class the next class by asking some of their questions.
Next we explored Sounding the Alarm from the Navigating Through Probability in Grades 9-12. In this investigation, I start with providing the scenario and asking what assumptions we would have to make for our simulation. This is always an interesting discussion. I try to push students to consider what they are assuming about each smoke alarm. Some questioning may be necessary to push them to consider the independence of each alarm.
The question of interest is to determine the probability that at least one alarm sounds. I have students consider how the situation could be simulated. In their groups students are very good at determining how to simulate a single alarm. Walking around I ask what they are thinking. Invariably I ask how they would now simulate all three alarms. After additional thought most groups determine a viable way to simulate all three alarms.
Ideas typically include using 4-sided die, spinners, card decks, and random numbers. I explain how students can ignore two sides of a 6-sided die and provide them with their devices of choice to simulate the situation. Again, some students struggle to simulate the results properly but a little guidance quickly gets them on track.
I had each student run 20 simulations and then pool their results within their groups. The idea is to get a probability based upon the entire classes simulation, approximately 500 simulations.
We got as far as completing the simulations and will analyze the data and compare against the theoretical results next class.
Visit my web site to see a student's perspective of the class and to see the lesson slides.
To connect with the idea of what randomness looks like and the Law of Large numbers the class started with a couple of quick assessments of understanding. Students voted on which heads-tails sequence was more likely. Roughly half thought the probability was the same and about 25% went to each individual sequence. The thoughts around the "hot hand" concept was divided roughly the same, half thought there was no such thing, 25% felt like the person was likely to continue with their streak, and 25% thought the streak would end. We discussed the reasoning and students were able to articulate the idea of the Law of Large Numbers.
The One Boy Family Planning activity comes from NTCM's Navigating Through Probability in Grades 9-12. I find the activities in this book to be enlightening and accessible to all levels of students.
I start the activity by asking students to read through the scenario and then develop a hypothesis as to what they think will happen given their current knowledge. We then briefly discuss the simulation aspect of the problem and I ask them to consider other ways that the situation could be simulated. Students came up with flipping a coin, using the random number generator, and mentioned that any device that allowed for two equal outcomes would work.
Students then proceeded to simulate the situation and record their results. Some groups will have misunderstanding of the simulation process so it's a good idea to walk around and make sure everyone is on track.
I ask students to hold off completing the second table as this usually causes problems. I actually have students create two columns: Total Children and Cummulative Children. I give a brief example to illustrate what this looks like. Again, most will get it but some may wander off track. I have students the total number of children and divide it by the number of households and ask what this represents. After a little thought the come up with the average children per household. I then explain how to use the second table to calculate their probabilities.
We discussed their expectations and surprises that came out of the investigation. Many students were somewhat surprised by the average children per household but they did anticipate that roughly half the houses would have a single child since the probability of having a boy was 50%.
I gave students a few minutes to collect their thoughts and summarize their thinking about randomness and to write two questions that they have. I will start class the next class by asking some of their questions.
Next we explored Sounding the Alarm from the Navigating Through Probability in Grades 9-12. In this investigation, I start with providing the scenario and asking what assumptions we would have to make for our simulation. This is always an interesting discussion. I try to push students to consider what they are assuming about each smoke alarm. Some questioning may be necessary to push them to consider the independence of each alarm.
The question of interest is to determine the probability that at least one alarm sounds. I have students consider how the situation could be simulated. In their groups students are very good at determining how to simulate a single alarm. Walking around I ask what they are thinking. Invariably I ask how they would now simulate all three alarms. After additional thought most groups determine a viable way to simulate all three alarms.
Ideas typically include using 4-sided die, spinners, card decks, and random numbers. I explain how students can ignore two sides of a 6-sided die and provide them with their devices of choice to simulate the situation. Again, some students struggle to simulate the results properly but a little guidance quickly gets them on track.
I had each student run 20 simulations and then pool their results within their groups. The idea is to get a probability based upon the entire classes simulation, approximately 500 simulations.
We got as far as completing the simulations and will analyze the data and compare against the theoretical results next class.
Visit my web site to see a student's perspective of the class and to see the lesson slides.
Tuesday, January 15, 2013
IPS - Day 4
Today we continued examining the idea of random events with the goal to re-inforce that what we think looks random may not be and that true random events may appear to be not random.
To start this class I had columns that represented months of the year. Students wrote the day of their birth under the appropriate month. I then asked students to comment on the data. Their were different observations about the distribution. I mentioned we would need data on births by month to see if our data fit the overall patterns or not. I then asked about the probability of having two birthdays on the same day. Most students thought that it was unusual. I asked if we increased the number of people to fifty or so what they thought would happen. About a third of the class responded that they thought we would get at least one match.
The fun part for this lesson is going to different classrooms and seeing if we got matches. I ask different teachers if we can interrupt their class for a few minutes and then have students in the class call out their birthday. Any time there was a match the person with the matching birthday would call out match. In visiting 3 classes we had matches of 3, 2, and 1. In the post discussion, students were curious about why the matches were so frequent. I told them we would look at how to calculate the probabilities as we progressed forward.
We then explored random sequences of heads and tails. By this time students were engaged in the idea of randomness and actively trying to imaging random coin tosses. The length of sequences (consecutive heads or tails) were tallied. We then used a calculator to simulate coin tosses and counted the length of random sequences. These typically showed a wider dispersion of sequence length and longer sequences. I wrote two sequences on the board, one reflecting the imagined results and one reflecting the random results. In looking at these sequences, students agreed the imagined results looked random while the random result looked fake while in reality it was the other way around.
We'll continue exploring the idea of randomness before we start to move into a more formal look at simulations.
For a student's perspective on the lesson and slides for the day, visit the class summary page on the course web site.
To start this class I had columns that represented months of the year. Students wrote the day of their birth under the appropriate month. I then asked students to comment on the data. Their were different observations about the distribution. I mentioned we would need data on births by month to see if our data fit the overall patterns or not. I then asked about the probability of having two birthdays on the same day. Most students thought that it was unusual. I asked if we increased the number of people to fifty or so what they thought would happen. About a third of the class responded that they thought we would get at least one match.
The fun part for this lesson is going to different classrooms and seeing if we got matches. I ask different teachers if we can interrupt their class for a few minutes and then have students in the class call out their birthday. Any time there was a match the person with the matching birthday would call out match. In visiting 3 classes we had matches of 3, 2, and 1. In the post discussion, students were curious about why the matches were so frequent. I told them we would look at how to calculate the probabilities as we progressed forward.
We then explored random sequences of heads and tails. By this time students were engaged in the idea of randomness and actively trying to imaging random coin tosses. The length of sequences (consecutive heads or tails) were tallied. We then used a calculator to simulate coin tosses and counted the length of random sequences. These typically showed a wider dispersion of sequence length and longer sequences. I wrote two sequences on the board, one reflecting the imagined results and one reflecting the random results. In looking at these sequences, students agreed the imagined results looked random while the random result looked fake while in reality it was the other way around.
We'll continue exploring the idea of randomness before we start to move into a more formal look at simulations.
For a student's perspective on the lesson and slides for the day, visit the class summary page on the course web site.
Discrete Math - Day 4
Today we looked at the problem of adding the squares of two consecutive triangular numbers. Students were having trouble see patterns. I wrote out the first five sums that students found so that all of the students were clear on the values being squared and summed. I then asked students to look for patterns and connections in the first nine sums. This provides students with three values that connect to each other. Most students still struggled with making connections but one student noticed that the sum of T12 and T22 was the same as T4 and that the sum of T22 and T32 was the same as T9. Someone asked if the sum of T32 and T42 equaled a square number and if it was T16. Students verified this and the next one by using the Gaussian summation formula to verify. The use of the Gaussian summation formula came from the students and it was encouraging to see them make use of this in reverse to validate the conjecture.
I briefly mentioned the idea of polygonal numbers and figurate numbers as being numbers that could be represented by dots arranged in the form of figures.
We then moved to pentagonal numbers. I asked students what a pentagonal number (see Wolfram Math - Pentagonal Number for more information) would look like and gave them a couple of minutes to think about how they might arrangement dots. Students were reluctant to share so I said that I'd get things going. I drew a single dot, labeled it as P1 and set it equal to 1. I said that now that I got things going someone could draw the next figure. Someone came up and drew five dots in the shape of a pentagon, labeled it as P2 and set it equal to five. A student had used their smartphone to google the next result of P3 = 12. We were a bit more challenged to draw the figure but finally had it drawn correctly. We also managed to get P4 drawn and get its value of 22.
The challenge was to find a general formula for Pn, the nth pentagonal number. I asked students to look for patterns and try to make connections. A student found a relationship between the value of n and the difference between Pn=1 and Pn. Specifically, he found Pn+1 = Pn + 3n + 1. This provided an opportunity to discuss recursive formulas. I related this back to the triangular numbers and the idea that most had found that Tn+1 = Tn + n + 1. I also mentioned that theoretically every recursive formula could be converted to a closed formula, such as Tn = n(n+1)/2. The next challenge was to find the closed formula for Pn.
Students have little experience converting non-linear relationships into formulas. In addition, even with linear expressions, many students are unsure and hesitant about creating a formula. I use this as a stepping off point to explore finite differences. It leads to many aha moments and provides a foundation for students to develop formulas from their table values.
Today was the first day that a student asked about when figurate numbers are used. I told them that the purpose of use working with these is for them to become better at reasoning and problem solving. These are unfamiliar items and they need to be able to work through the uncertainty, making connections to things they already know and to find patterns that they can explore and try to make sense of in the context of the problem. They seemed comfortable with the response. What they will come to realize is that this material is some of the easier material they will face and that building their confidence and abilities to explore, reason, and persist in problem solving now will serve them well as we dive into more complex problems.
Visit the class summary for a student's perspective of the day's lesson.
I briefly mentioned the idea of polygonal numbers and figurate numbers as being numbers that could be represented by dots arranged in the form of figures.
We then moved to pentagonal numbers. I asked students what a pentagonal number (see Wolfram Math - Pentagonal Number for more information) would look like and gave them a couple of minutes to think about how they might arrangement dots. Students were reluctant to share so I said that I'd get things going. I drew a single dot, labeled it as P1 and set it equal to 1. I said that now that I got things going someone could draw the next figure. Someone came up and drew five dots in the shape of a pentagon, labeled it as P2 and set it equal to five. A student had used their smartphone to google the next result of P3 = 12. We were a bit more challenged to draw the figure but finally had it drawn correctly. We also managed to get P4 drawn and get its value of 22.
The challenge was to find a general formula for Pn, the nth pentagonal number. I asked students to look for patterns and try to make connections. A student found a relationship between the value of n and the difference between Pn=1 and Pn. Specifically, he found Pn+1 = Pn + 3n + 1. This provided an opportunity to discuss recursive formulas. I related this back to the triangular numbers and the idea that most had found that Tn+1 = Tn + n + 1. I also mentioned that theoretically every recursive formula could be converted to a closed formula, such as Tn = n(n+1)/2. The next challenge was to find the closed formula for Pn.
Students have little experience converting non-linear relationships into formulas. In addition, even with linear expressions, many students are unsure and hesitant about creating a formula. I use this as a stepping off point to explore finite differences. It leads to many aha moments and provides a foundation for students to develop formulas from their table values.
Today was the first day that a student asked about when figurate numbers are used. I told them that the purpose of use working with these is for them to become better at reasoning and problem solving. These are unfamiliar items and they need to be able to work through the uncertainty, making connections to things they already know and to find patterns that they can explore and try to make sense of in the context of the problem. They seemed comfortable with the response. What they will come to realize is that this material is some of the easier material they will face and that building their confidence and abilities to explore, reason, and persist in problem solving now will serve them well as we dive into more complex problems.
Visit the class summary for a student's perspective of the day's lesson.
Monday, January 14, 2013
IPS - Day 3
Today we wrapped up our look into theoretical and experimental probability. We opened with a discussion of the similarities and differences between the rolling die and tossing hair clip experiments. This lead to a discussion of the "Law of Large Numbers" and the fallacious "Law of Averages." Examples were provided, such as an athlete having a hot hand or being due to score. An explicit connection was made between experimental and theoretical probability. For students who had pre-calculus or calculus, you can relate this to a limiting process. A few additional examples were provided to help reinforce the idea.
Next, another experiment was conducted by tossing a coin 100 times. As before, students were asked to consider what they thought would happen. The number of heads minus the number of tails becomes the response variable that is measured. I had students put these values up on the board along with the longest streak of heads or tails that was tossed. Most students expected to see heads and tails within 4 or 5 of each other. They were surprised to see one student had tossed 38 heads and 62 tails. Another student had 58 heads and 42 tails. These exceeded what students thought might happen. The length of streaks was also surprising to students. Many thought 3-5 would be the length of the longest streak but there were many that were 10 or more, including one that was 15 and another that was 13.
This was an opportunity to introduce the idea that the data we gathered was random data and that randomness often looks quite different from what our mind conceives. I used this as an opportunity to mention that statistics looks at random data that we generate and compares it to our understanding of what random data should look like. If the two mesh then everything is fine, but if the two don't mesh we need to understand what is at play to cause the difference.
We finished with playing a coin tossing game. Again, students set down expectations before playing. They were surprised at the percentage of games that were won by a 4-1 split versus a 3-2 split.
For reflection, students considered connections between theoretical probability, experimental probability, and the "Law of Large Numbers." They were also asked to state the law in their own words and provide an example.
Visit the course notes for the day to read a student's perspective about the class and see the day's slides.
Next, another experiment was conducted by tossing a coin 100 times. As before, students were asked to consider what they thought would happen. The number of heads minus the number of tails becomes the response variable that is measured. I had students put these values up on the board along with the longest streak of heads or tails that was tossed. Most students expected to see heads and tails within 4 or 5 of each other. They were surprised to see one student had tossed 38 heads and 62 tails. Another student had 58 heads and 42 tails. These exceeded what students thought might happen. The length of streaks was also surprising to students. Many thought 3-5 would be the length of the longest streak but there were many that were 10 or more, including one that was 15 and another that was 13.
This was an opportunity to introduce the idea that the data we gathered was random data and that randomness often looks quite different from what our mind conceives. I used this as an opportunity to mention that statistics looks at random data that we generate and compares it to our understanding of what random data should look like. If the two mesh then everything is fine, but if the two don't mesh we need to understand what is at play to cause the difference.
We finished with playing a coin tossing game. Again, students set down expectations before playing. They were surprised at the percentage of games that were won by a 4-1 split versus a 3-2 split.
For reflection, students considered connections between theoretical probability, experimental probability, and the "Law of Large Numbers." They were also asked to state the law in their own words and provide an example.
Visit the course notes for the day to read a student's perspective about the class and see the day's slides.
Discrete Math - Day 3
We started the day summarizing what was learned about triangular and square numbers from last class. Since this was a Monday morning class I thought it would be a good idea to have students re-engage in their thinking and notes before tackling the next set of problems.
We wrote out the formulas for Tn and Sn, the values of the nth triangular and square numbers, respectively. It was interesting that students struggled with the formula for Sn = n2, since we didn't explicitly discuss this last week. They were much more comfortable reciting the Gaussian summation formula for triangular numbers, Tn = n(n+1)/2.
Students tackled the next three problems related to triangular numbers. The purpose of these are to get students comfortable working through problems. I warned students that part e was a tougher pattern to see. Students made slow progress on these but did identify the patterns for d and f. For part d, we have
Tn-1 + Tn = n2.
For part f we have T2n-1 + T2n = n3.
After having students present results I wrote out the associated formulas for both problems. There wasn't much progress on part e, so I asked students to write out the first eight terms and to look for a pattern.
Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.
We'll work with these tomorrow, along with looking at pentagonal numbers. I intend on the pentagonal numbers to be the first portfolio problem.
We wrote out the formulas for Tn and Sn, the values of the nth triangular and square numbers, respectively. It was interesting that students struggled with the formula for Sn = n2, since we didn't explicitly discuss this last week. They were much more comfortable reciting the Gaussian summation formula for triangular numbers, Tn = n(n+1)/2.
Students tackled the next three problems related to triangular numbers. The purpose of these are to get students comfortable working through problems. I warned students that part e was a tougher pattern to see. Students made slow progress on these but did identify the patterns for d and f. For part d, we have
Tn-1 + Tn = n2.
For part f we have T2n-1 + T2n = n3.
After having students present results I wrote out the associated formulas for both problems. There wasn't much progress on part e, so I asked students to write out the first eight terms and to look for a pattern.
Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.
We'll work with these tomorrow, along with looking at pentagonal numbers. I intend on the pentagonal numbers to be the first portfolio problem.
Friday, January 11, 2013
IPS - Day 2
The next two days focuses on understanding theoretical and experimental probability. We were actually able to start this lesson on the first day and had students complete the 40 die rolls. They completed the results and calculations as homework.
I then had students record their graph results on the board. I think that next year I'll pass around a graph that each group can plot on and then project the results. This will prevent students from waiting around while graphs are plotted on the board.
We discussed what they saw on the graphs (a convergence of the results) and the asked what would happen if we continued rolling the die more and more. I accumulated total counts for all the groups to calculate a value off a larger number.We compared this result to the theoretical result.
We then moved to tossing a hair clip which does not have equal probability. You can use any six-sided, asymmetrical object. I ask students to develop a hypothesis, their guess based on their current understand, as to what the probability might be. We then conducted the experiment and saw comparable graph convergence. We calculated a probability using the cumulative class data. This becomes our best estimate as to what the theoretical value would be. I also convey that we would need millions of tosses to hone in on a value.
We concluded class with students recording their thoughts about the two experiments: similarities, differences, insights, surprises, and understandings.
All of this is building to understanding the connection between experimental and theoretical probability and the Law of Large Numbers.
Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.
I then had students record their graph results on the board. I think that next year I'll pass around a graph that each group can plot on and then project the results. This will prevent students from waiting around while graphs are plotted on the board.
We discussed what they saw on the graphs (a convergence of the results) and the asked what would happen if we continued rolling the die more and more. I accumulated total counts for all the groups to calculate a value off a larger number.We compared this result to the theoretical result.
We then moved to tossing a hair clip which does not have equal probability. You can use any six-sided, asymmetrical object. I ask students to develop a hypothesis, their guess based on their current understand, as to what the probability might be. We then conducted the experiment and saw comparable graph convergence. We calculated a probability using the cumulative class data. This becomes our best estimate as to what the theoretical value would be. I also convey that we would need millions of tosses to hone in on a value.
We concluded class with students recording their thoughts about the two experiments: similarities, differences, insights, surprises, and understandings.
All of this is building to understanding the connection between experimental and theoretical probability and the Law of Large Numbers.
Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.
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